UK Nonlinear News, February 1999

Dynamical Systems and Ergodic Theory

By M. Pollicott and M. Yuri

Reviewed by Jaroslav Stark

Paperback ISBN: 0 521 57599 0, #14.95 (#13.64 from http://www.amazon.co.uk)
Hardback ISBN: 0 521 57294 0, #40.00

Topological dynamics and ergodic theory are some of the oldest branches of nonlinear dynamics. Despite this, they are on the whole less well known than say hyperbolic dynamics or bifurcation theory, particularly amongst those working on the more applied side of the subject. As a result there are far fewer introductory textbooks on either subject, compared to the wealth of titles taking a geometric or applied approach. The only volume that immediately comes to mind is Introduction to the Modern Theory of Dynamical Systems by A. Katok and B. Hasselblatt in the CUP Encyclopaedia of Mathematics and its Applications series. This is comprehensive, very clear and well written, but if only because of its bulk (over 800 pages) also rather intimidating. It is hardly the thing to give to a starting graduate student if all they require is a passing familiarity with the two subjects.

This situation is somewhat unfortunate since many fundamental concepts in daily use in applied dynamics depend on their definition on either ergodic theory (e.g. Liapunov exponents) or topological dynamics (e.g. attractors). The appearance of the much slimmer (190 page) volume under review is thus very welcome especially when its stated aim is

This book is essentially a self-contained introduction to topological dynamics and ergodic theory. It is divided into a number of relatively short chapters with the intention that each may be used as a component of a lecture course tailored to the particular audience. Parts of the book are suitable for a final year undergraduate course or for a master’s level course.

Furthermore, the book’s first author is undoubtedly the UK’s foremost young ergodicist, and his previous book (Lectures on Ergodic Theory and Pesin Theory on Compact Manifolds, CUP, 1993) was by far the most readable account of the Multiplicative Ergodic Theorem and of Pesin Theory (which are the branches of ergodic theory dealing with Liapunov exponents and their consequences).

Overall, the volume achieves its goals well. It covers a broad range of topics clearly and succinctly. Each chapter is brief and to the point, and on the whole the proofs of various theorems are well presented and easy to follow. There are occasional typographical errors, and some rather wayward notation (eg in Chapter 3 the symbol h alternates between topological entropy and topological conjugacy until in the last few equations it assumes both roles simultaneously), but on the whole this is a minor problem. For such a short book it includes a surprising breadth of topics, and I was delighted to find results as diverse as Denjoy’s Theorem on circle maps, Szemerdi’s Theorem on arithmetic progressions and the Poincaré-Birkhoff Theorem on area-preserving maps. There is thus much material here to interest and stimulate the reader.

Unfortunately, the brevity and self-contained nature of each chapter also has some drawbacks. Firstly, the volume requires a significant level of mathematical sophistication, and assumes quite an advanced level of prior knowledge. I think that most new graduate students would find it difficult going, and it would only really be useful in a second term graduate course. Even then, it would be rather inaccessible to students without a fairly pure mathematical background. Secondly, most of the concepts introduced in the book are lacking in motivation. Thus, for example, the reader is given the definition of topological conjugacy, but actually provided with little insight into why this is such an important concept. Finally, there is a lack of any clear theme running through the book. Since most chapters are largely self-contained, the volume gives more of an impression of a selection of interesting results, than a coherent development of ergodic theory or topological dynamics. This is not much of a problem for someone who already has some kind of overview of these subjects, but again makes the book less useful as an introductory text.

Overall, I thus think that this volume will have limited appeal as an introduction to ergodic theory or topological dynamics for the general nonlinear community. However, I thoroughly recommend it to anyone of has some knowledge of the subject matter and wants a concise and well presented reference for more advanced concepts.

Dr. Jaroslav Stark,
Monday, January 18, 1999.

UK Nonlinear News thanks Cambridge University Press for providing a copy of this book for review.


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